Lemma 110.68.1. Let R \to R' and R \to A be ring maps. In general there does not exist a functor T : D(A) \to D(A \otimes _ R R') of triangulated categories such that an A-module M gives an object T(M) of D(A \otimes _ R R') which maps to M \otimes _ R^\mathbf {L} R' under the map D(A \otimes _ R R') \to D(R').
110.68 Derived base change
Let R \to R' be a ring map. In More on Algebra, Section 15.60 we construct a derived base change functor - \otimes _ R^\mathbf {L} R' : D(R) \to D(R'). Next, let R \to A be a second ring map. Picture
Given an A-module M the tensor product M \otimes _ R R' is a A \otimes _ R R'-module, i.e., an A'-module. For the ring map A \to A' there is a derived functor
but this functor does not agree with - \otimes _ R^\mathbf {L} R' in general. More precisely, for K \in D(A) the canonical map
in D(R') constructed in More on Algebra, Equation (15.61.0.1) isn't an isomorphism in general. Thus one may wonder if there exists a “derived base change functor” T : D(A) \to D(A'), i.e., a functor such that T(K) maps to K \otimes _ R^\mathbf {L} R' in D(R'). In this section we show it does not exist in general.
Let k be a field. Set R = k[x, y]. Set R' = R/(xy) and A = R/(x^2). The object A \otimes _ R^\mathbf {L} R' in D(R') is represented by
and we have H^0(A \otimes _ R^\mathbf {L} R') = A \otimes _ R R'. We claim that there does not exist an object E of D(A \otimes _ R R') mapping to A \otimes _ R^\mathbf {L} R' in D(R'). Namely, for such an E the module H^0(E) would be free, hence E would decompose as H^0(E)[0] \oplus H^{-1}(E)[1]. But it is easy to see that A \otimes _ R^\mathbf {L} R' is not isomorphic to the sum of its cohomology groups in D(R').
Proof. See discussion above. \square
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